Dynamical Systems in Neuroscience:

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40 Electrophysiology of NeuronsFigure 2.14: Studies of spike-generation mechanism in “giant squid axons” won AlanHodgkin and Andrew Huxley the 1963 Nobel prize for physiology or medicine (sharedwith Sir John Eccles); see also Fig. 4.1 in Keener and Sneyd (1998).Since we are interested in geometrical and qualitative methods of analysis of neuronalmodels, we assume that all variables and parameters have appropriate scalesand dimensions, but we do not explicitly state them. An exception is the membranepotential V , whose mV scale is stated in every figure.2.3.2 Action PotentialRecall that when V = V rest , which is 0 mV in the Hodgkin-Huxley model, all inwardand outward currents balance each other so the net current is zero, as in Fig. 2.15.The rest state is stable: A small pulse of current applied via I(t) produces a smallpositive perturbation of the membrane potential (depolarization), which results in asmall net current that drives V back to rest (repolarization). However, an intermediatesize pulse of current produces a perturbation that is amplified significantly becausemembrane conductances depend on V . Such a non-linear amplification causes V todeviate considerably from V rest — a phenomenon referred to as an action potential orspike.In Fig. 2.15 we show a typical time course of an action potential in the Hodgkin-Huxley system. Strong depolarization increases activation variables m and n and decreasesinactivation variable h. Since τ m (V ) is relatively small, variable m is relativelyfast. Fast activation of Na + conductance drives V toward E Na , resulting in furtherdepolarization and further activation of g Na . This positive feedback loop, depicted inFig. 2.16 results in the upstroke of V . While V moves toward E Na , the slower gatingvariables catch up. Variable h → 0, causing inactivation of Na + current, and variablen → 1, causing slow activation of outward K + current. The latter and the leak current
Electrophysiology of Neurons 41E Na10050E L0E KMembrane voltage (mV)smalldepolarizationRestRepolarizationExcited(regenerative)largedepolarizationUpstrokeActionpotential(spike)RepolarizationAbsoluterefractorydepolarizationresthyperpolarizationAfter-hyperpolarizationRelativerefractoryV(t)10.50Gating variablesactivationh(t)n(t)m(t)inactivationdeactivationdeinactivation20Conductances (mS/cm2)g (t)Nag (t)K05000-500Currents( µ A/cm 2)I Na+I K+ILI (t)KI (t)Na200Applied current( µ A/cm 2)0 2 4 6 8 10 12 14 16 18 20time (ms)Figure 2.15: Action potential in the Hodgkin-Huxley model.I(t)
Page 1: Eugene M. IzhikevichThe Neuroscienc
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Page 10 and 11: xPrefaceversely, cells having quite
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Page 16 and 17: 6 Introduction1.1.3 Why are neurons
Page 18 and 19: 8 Introductionconcepts of dynamical
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Page 24 and 25: 14 Introductionco-existence of rest
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Page 30 and 31: 20 Introductionwith coupled neurons
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90 One-Dimensional Systems10.80.60.
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92 One-Dimensional Systems
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94 Two-Dimensional SystemsI Na,pneu
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96 Two-Dimensional Systemsdefines a
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98 Two-Dimensional Systems(f(x(t),y
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100 Two-Dimensional Systems0.70acti
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102 Two-Dimensional Systems0.70.6K
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104 Two-Dimensional Systemsy1.510.5
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106 Two-Dimensional SystemsFor exam
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110 Two-Dimensional Systemsv 2 v 2v
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112 Two-Dimensional SystemsV-nullcl
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114 Two-Dimensional SystemsV-nullcl
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116 Two-Dimensional Systemsheterocl
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118 Two-Dimensional Systems0.60.5n-
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120 Two-Dimensional Systemseigenval
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122 Two-Dimensional SystemsK + acti
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124 Two-Dimensional Systemsexcitati
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126 Two-Dimensional SystemsReview o
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128 Two-Dimensional SystemsabcdFigu
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130 Two-Dimensional SystemsFigure 4
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132 Two-Dimensional Systems
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134 Conductance-Based Models• If
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136 Conductance-Based Modelsinward(
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138 Conductance-Based Models1I=01I=
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140 Conductance-Based Modelsleak cu
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142 Conductance-Based Modelshyperpo
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144 Conductance-Based Models0I=0ina
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146 Conductance-Based Models0I=9.75
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148 Conductance-Based Models1I=651I
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150 Conductance-Based Modelshave co
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152 Conductance-Based Models1 sec10
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154 Conductance-Based Modelsvoltage
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156 Conductance-Based Models1h=0.89
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158 Conductance-Based Models5.2.2 E
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160 Conductance-Based Models1recove
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162 Conductance-Based Modelscurrent
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164 Conductance-Based Modelsmodels
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166 Conductance-Based Models
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168 Bifurcationssaddle-node bifurca
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170 Bifurcationssaddle-node bifurca
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172 Bifurcations0.5v 2V-nullclineK
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174 BifurcationsBoth types of the b
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176 BifurcationsV-nullcline0.5K + g
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178 Bifurcationsrstable limit cycle
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180 BifurcationsnVstable limit cycl
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¡ ¡¡ ¡¡ ¡ ¡¡ ¡ ¡182 Bifur
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186 Bifurcationsmembrane potential,
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188 BifurcationsBifurcation of a li
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190 Bifurcationsstableunstablelimit
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192 Bifurcationsamplitude (max-min)
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194 Bifurcationshomoclinic orbithom
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196 Bifurcations0.80.60.4stable lim
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198 Bifurcationsfrequency (Hz)40030
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200 BifurcationsSaddle-Focus Homocl
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202 Bifurcationsc 1c 2xFigure 6.34:
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204 BifurcationseigenvaluesHopffold
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206 Bifurcationsfast nullclineslow
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208 Bifurcationswith fast and slow
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210 BifurcationsSupercritical Andro
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212 BifurcationsK + conductance tim
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214 BifurcationsIn contrast, if the
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216 Bifurcationsinvariant circlesad
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218 Bifurcationsbifurcationssaddle-
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220 BifurcationsExercises1. (Transc
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222 Bifurcationsv 2v 11a-11-1Figure
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224 Bifurcations19. [M.S.] A leaky
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226 Excitabilityspike?spikerestrest
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228 ExcitabilityAlternatively, the
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230 Excitability50 ms 20 mV 1 ms100
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232 ExcitabilityClass 3 excitable n
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234 Excitability0.20.150.10.05I 1I
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236 Excitabilitysaddle-node bifurca
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238 Excitability(a) resting spiking
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240 Excitabilityproperties integrat
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242 ExcitabilityThe existence of fa
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244 Excitability1coincidencedetecti
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246 Excitabilityspikeexcitatory pul
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248 Excitability(g)9 ms(f)(e)(d)(c)
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250 Excitabilitysquid axonmodel0 mV
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252 Excitability(a) integrator(b) r
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254 Excitability(a) integrator(b) r
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256 Excitabilitymembrane potential,
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258 Excitability0.1saddle-node bifu
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260 Excitability100 ms10 mVoscillat
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262 ExcitabilityBogdanov-Takens bif
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264 Excitabilitya pulse of current.
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266 Excitabilitymembrane potential
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268 Excitability(a)150100I K(M)(b)3
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270 Excitability50 mV300 msFigure 7
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272 Excitability20 mVADP100 msAHP-6
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274 ExcitabilityK + activation gate
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276 ExcitabilityBibliographical Not
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278 Excitability7. Show that the re
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280 Simple Modelsspikemembrane pote
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282 Simple Models1thresholdyz reset
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284 Simple Models1v reset =|b| 1/2b
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286 Simple Modelsbe an integrator o
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288 Simple Modelsintegrate-and-fire
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290 Simple Models(A) tonic spiking(
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292 Simple Modelsgeneral algorithm
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294 Simple Modelscha, cha — real
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296 Simple Modelslayer 5 neuronsimp
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298 Simple Modelsb=-240200saddle-no
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300 Simple Models(a)simple modellay
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306 Simple Models(a)74 5 6 3210(b)C
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308 Simple Modelschattering neuron
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310 Simple ModelsLTS neuron (in vit
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312 Simple ModelsFS neuron (in vitr
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314 Simple Models(1) fast oscillati
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316 Simple Modelsthrough an appropr
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318 Simple Modelsthey are able to g
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320 Simple Modelsv r = −80 mV, an
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322 Simple Modelsshow in Fig. 7.36.
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324 Simple ModelsBibliographical No
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326 Simple ModelsExercises1. (Integ
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328 Simple Models17. [M.S.] Analyze
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330 Simple Modelsa35 mV350 msc-NAC
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332 Simple Modelsrat RTN neuronsimp
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334 Simple ModelsFigure 8.34: Class
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336 Simple Modelsspiny neuronlatenc
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338 Simple Models(a)stellate cellof
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340 Simple Modelsrat's mitral cell
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342 Bursting(a) cortical chattering
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344 Burstingmembranepotential (mV)-
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346 Burstingvoltage-gatedCa2+-gated
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348 Burstingslow dynamicsneuronvolt
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350 Burstingslow inactivation of in
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352 Bursting9.2.1 Fast-slow burster
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354 Burstingn-nullclinen slow =-0.0
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356 Bursting0maxmembrane potential,
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358 Bursting0membrane potential, V
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360 BurstingI=0I=4.5425 ms 25 mVI=5
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362 Burstingmembrane potential, V (
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364 Bursting9.3 ClassificationIn Fi
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366 Burstingbifurcation of spiking
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368 BurstingFigure 9.26: Putative
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370 Bursting108spikingslow variable
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372 Bursting(a)membrane potential,
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374 Burstingdepending on the type o
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376 BurstingsubcriticalAndronov-Hop
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378 Bursting(a)membrane potential,
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spiking380 Burstingfoldbifurcations
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382 BurstingspikingsupercriticalAnd
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384 Burstingaction potentials cut2
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386 Bursting9.4.3 BistabilitySuppos
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388 BurstingFigure 9.49: The instan
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esting390 Burstingspikesynchronizat
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392 BurstingReview of Important Con
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394 BurstingspikingrestingFigure 9.
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396 Bursting0-10membrane potential,
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398 Bursting0membrane potential, V
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400 BurstingFigure 9.61: A cycle-cy
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402 Bursting28. [Ph.D.] Develop an
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404 Synchronization (see www.izhike
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408 Solutions to Exercises, Chap. 3
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442 Referencesterneurons mediated b
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444 ReferencesDickson C.T., Magistr
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446 ReferencesGuckenheimer J. (1975
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448 Referencestional Journal of Bif
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450 ReferencesMarkram H, Toledo-Rod
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452 ReferencesRosenblum M.G. and Pi
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454 ReferencesTuckwell H.C. (1988)
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456 References9
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